Prakash, Anand

Srivastava, Pramila

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaAnand Prakash; Pramila SrivastavaSomewhat continuity and some other generalisations of continuityMathematica Slovaca, Vol. 27 (1977), No. 3, 243--248Persistent URL: http://dml.cz/dmlcz/136148Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slovaca 27,1977, No. 3, 243—248 SOMEWHAT CONTINUITY AND SOME OTHER GENERALISATIONS OF CONTINUITY ANAND PRAKASH—P. SRIVASTAVA 1. Introduction. One of the notions generalising the notion of continuity is the almost continuity. It has been defined differently by Stallings [23], Frolik [3], Husain [6], and Singal and Singal [21]. Long and McGehene Jr. [10], Deb [1], Long and Carnahan [9], and Herrington, Hunsaker, Lindgren and Naimpally [5] have shown that the four definitions of the almost continuity are independent of one another. Also Singal and Singal [21], Deb [1], and Papp [17] gave examples to show that the class of the ^-continuous mappings contained properly the class of the almost continuous mappings of Singal and Singal, and was itself contained properly in the class of the weakly continuous mappings. In 1971, Gentry and Hoyle [4] introduced another weaker form of continuity called the "somewhat continuity". Frolik [3], Neubrunnova [14], Neubrunn [15] studied interrelations between the somewhat continuity; the quasi continuity, the semi continuity and some types of the almost continuity. In the present paper we further investigate the relationship of the somewhat continuity with other types of the almost continuity, the ^-continuity and weak continuity. Also the authors in their papers [19, 20] gave conditions for which the almost continuity of Singal and Singal, and the weak continuity, respectively, imply the almost continuity of Husain. Now we shall obtain conditions (Theorem 6) under which the almost continuity of Husain would imply the almost continuity of Singal and Singal and hence ^-continuity. Our Theorem 5 extends the results of a theorem of Wilansky [24, Theorem 3.5] for the class of continuous functions to the class of weakly continuous functions. 2. Notations and definitions. Let A be a subset of a topological space X, The interior of A, the closure of A, and the complement of A in X are denoted by lnt(A), CI (A), and (X—A), respectively. A is semi-open if there exists an open set O in X such that O c A cz Cl(O). S • O • (X) denotes the class of all semi-open sets in X. A is regular open if A = Int(Cl(_4)). 243 Definition 1 [Stallings, 23]. A mapping f: X-* Yis said to be almost continuous, if for every open set W containing the graph off, there exists a continuous mapping g: X-* Y such that the graph of g is a subset of W. Definition 2 [Frolik, 3]. A mapping f: X-* Yis said to be almost continuous if for every open subset Vof Y, f~l(V)czC\(lnt(rl(V))). Definition 3 [Husain, 6]. A function f: X-+ Yis said to be almost continuous at x eXiffor each open Va Ycontaining f(x), C\(f~\ V)) is a neighborhood ofx. If f is almost continuous at each point of X, then f is called almost continuous. Definition 4 [Singal and Singal, 21]. A mapping f:X-+ Y is said to be almost continuous at a point xeX, if for every neighborhood M of f(x), there is a neighborhood N of x such that f(N)czlnt(C\)(M)). f is said to be almost continuous if it is almost continuous at each point x of X. Definition 5 [7]. A mapping f: AT—> Yis said to be weakly continuous if for each point xeX and each neighborhood V of f(x), there exists a neighborhood U of x such thatf(U)czC\(V). Definition 6 [8]. A mapping f: X—> X* is semi-continuous iff for O* open in X*, rl(0*)eSO(X). Definition 7 [2]. A mapping f: X-» Yis said to be 6-continuous if for each point xeX and each neighborhood V of f(x) there is a neighborhood U of x such that f(C\(U))czC\(V). Definition 8 [11, 12]. A mapping f:X^> Y is said to be quasicontinuous if for every point xeX and for every open set V containing x and every open set U containing f(x) there exists a non-empty open set W such that Wcz V and f(W)czU. Definition 9 [4]. Let (X, STX) and (Y, 3~2) be topological spaces. A function f: (X, J",)—>(y, :T2) is said to be somewhat continuous provided that if U e 5"2 and rl(U)±0, then there is a Ve3~x such that V±0 and Vczrl(U). Definition 10 [24]. / / / : X-^> Yis a one to one mapping, then it is almost open iff for every open G^Y, rl(C\(G))czC\(rl(G)). Definition 11 [22]. A space X is almost regular if for each point xeX and each regular open set V containing x, there exists a regular-open set U such that xeUaC\(U)czV. 244 3. Somewhat continuity and other weaker forms of continuity. Theorem 1. The almost continuity of Stallings and the somewhat continuity are two independent notions. Proof. We prove it by giving two examples: Example 1. Consider the mapping f:(X, 3'1)->(X, 3~2), where X={a,b}, 3~x = {0, {a},X}, 3~2 = {0,X, {a}, {b}}, and f{a) = a, f(b) = b. Then / is almost continuous in the sense of Stallings [13, Ex. 3.1], but / is not somewhat continuous because U={b}e 3~2 and /"'(£/) = {b} 40, but no Ve3~x, V40 is such that Vczf'\U). This shows that the almost continuity of Stallings does not imply the somewhat continuity. Example 2. Let X= [0, 1], Y= {a, b, c}, 3~x is the usual topology for X, and 3~2 = {Y, 0, {c}, {a}, {a, c}}. Consider the mapping / : (X, 3~X)-*(Y, 3'2) defined by f(x) = A, if 0=5* =£; f(x) = c, if -2<Л:Í£І Then / is not almost continuous in the sense of Stallings, but / is somewhat continuous. From this example we assert that the somewhat continuity does not imply the almost continuity of Stallings. Theorem 2. The almost continuity of Singal and Singal and the somewhat continuity are two independent notions. Proof. Consider the following example. Example 3. Let /': (R, 3~X)-+(R, 3~2) be the identity mapping, where R is the set of reals, 3~x is the usual topology on R, 3~2 the co-countable topology. Then i is almost continuous in the sense of Singal and Singal, but it is not somewhat continuous. Thus the almost continuity of Singal and Singal does not imply the somewhat continuity. Example 4. Let (X, 3~x) and (Y, 3~2) be topological spaces, where X={a, b, c, d}, Y={p,q,r} 3-x = {0, X, {a, c}, {d}, {c}, {c, d}, {a, c, d}} 3~2 = {0,Y,{r},{q},{r,q}}. Define f:X-*Y by f(a)=f(d) = p, f(b)=f(c) = r. Then / is somewhat continuous but not almost continuous in the sense of Singal and Singal. So the somewhat continuity does not imply the almost continuity of Singal and Singal. 245 Theorem 3. The almost continuity of Husain and the somewhat continuity are two independent notions. Proof. We give the following example: Example 5. Consider the mapping f:(R, 5r,)—>(Y, J2), Where /?=set of reals, Y={a,b}, &l = {AczR\(R-A) is finite}, SF2 = {y, 0, {a}}, end / is defined by f(x) = a if x is irrational; f(x) — b if x is rational. Then / is almost continuous in the sense of Husain, but / is not somewhat continuous. Thus the almost continuity of Husain does not imply the somewhat continuity. Next consider Example 4. The mapping defined there is somewhat continuous but not almost continuous in the sense of Husain. So the somewhat continuity does not imply the almost continuity of Husain. The referee has brought to the notice of the authors that the above theorem has also been proved with the help of different examples in the thesis of A. Neubrunnova. Theorem 4. The somewhat continuity is independent of the weak continuity and the 6-continuity. Proof. The fact that neither the 6-continuity nor the weak continuity implies the somewhat continuity follows from Example 3. The converse is prove'd by the mapping defined in Example 4 which is somewhat continuous but is neither 0-continuous nor weakly continuous. We make use of the following lemma in our next theorem. Lemma 1 [16, 18]. Letf: X—> Ybe a weakly continuous function. Then for each open Vcz y, C\(f-l(V))czf-\C\(V)). Theorem 5. Let f: X—> Y be one to one and weakly continuous. Then the following are equivalent. (1) f is almost open. (2) For every open set Va Y, f~l(C\( V)) = Cl(/_1( V)). Proof. From Definition 10 and Lemma 1 we get the required result. This improves upon a result of Wilansky [24, Theorem 3.5] by replacing the continuity of / by its weak continuity. For our next theorem we need the following lemmas. Lemma 2 [18, Theorem 5.5; 16]. Iff: X-± Yis almost continuous in the sense of Husain such that for each open Vc Y, C\(f~l( V))c/-!(C1( V)). Then f is weakly continuous. Lemma 3 [1]. If f:X-^> Y is a weakly continuous mapping and Y is almost 246 regular, then f is almost continuous in the sense of Singal and Singal (hence 0-continuous). Theorem 6. Let Y be an almost regular space. Let f: X—> Y be a mapping such that for every open set Vcz Y, C\{f~\V))czf~l{C\{ V)). If f is almost continuous in the sense of Husain, then f is almost continuous in the sense of Singal and Singal (and hence 6-continuous). Proof. By combining Lemma 2 and Lemma 3 we get the assertion of the theorem. The authors are grateful to Anna Neubrunnova for her kind suggestions. REFERENCES [1] MAMATA DEB: A study of certain types of mappings on topological spaces. Ph.D. Thesis, Delhi University, 1971. [2] FOMIN, S. V.: Extension of topological spaces. Ann. Math. 44, 1943, 471—480. [3] FROLIK, Z.: Remarks concerning the invariance of Baire spaces under mappings. Czech. Math. J. 11(86), 1961, 381—385. [4] GENTRY, KARL R.—HOYLE III, HUGHES B.: Somewhat continuous functions. Czech. Math. J. 21(96), 1971, 5—12. [5] HERRINGTON, L. L.~-HUNSAKER, W. N.-LINDGREN, W. F.—NAIMPALLY, S. A.: A counter example concerning almost continuous functions. Proc. Amer. Math. Soc. 43, 1974, 475. [6] HUSAIN, T.: Almost continuous mappings. Prace Mat. 10, 1966, 1—7. [7] LEVINE, N.: A decomposition of continuity in topological spaces. Amer. Math. Monthly 68, 1961, 44—46. [8] LEVINE, N.: Semi-open sets and semi-continuity in topological spaces. Amer. Math. Monthly 70, 1963, 36—41. [9] LONG, P. E.—CARNAHAN, D. A.: Comparing almost continuous functions. Proc. Amer. Math. Soc. 38, 1973, 413—418. [10] LONG, P. E.—McGEHEE Jr., E. E.: Properties of almost continuous functions. Proc. Amer. Math. Soc. 24, 1970, 175—180. [11] MARCUS, S.: Sur les fonctions quasicontinues au sens de S. Kempisty. Colloq. Math. Wroclaw 8, 1961, 47—53. [12] MARTIN, N. F. G.: Quasi continuous functions on product spaces. Duke Math. J. 28, 1961, 39—44. [13] NAIMPALLY, S. A.: Graph topology for function spaces. Trans. Amer. Math. Soc. 123, 1966, 267—272. [14] NEUBRUNNOVA, A.: On certain generalisations of the notion of continuity. Mat. Cas. 23, 1973, 374—380. [15] NEUBRUNN, T.: On semihomeomorphism and related mappings. Acta F.R.N. Univ. Comen. Math. 1976 (to appear). [16] TAKASHI NOIRI: On weakly continuous mappings. Proc. Amer. Math. Soc. 46, 1974, 120—124. [17] PAPP, F. J.: Almost continuous implies 0-continuous. Notices Amer. Math. SoCj_1973. [18] AN AND PRAKASH: A study of almost continuity and some other weaker forms of continuity in topological spaces. Ph.D. Thesis, Banaras Hindu University, 1974. 247 [19] ANAND PRAKASH—SRIVASTAVA, P.: A note on almost continuous functions. J. Math. Soc, Banaras Hindu University Varanasi 1969, 23—24. [20] ANAND PRAKASH—SRIVASTAVA, P.: A note on weak continuity and almost continuity. The Yokohama Math. J. 20, 1972, 131—134. [21] SINGAL, M. K—ASHA RANI SINGAL: Almost continuous mappings. Yokohama Math. J. 16, 1968, 63—73. [22] SINGAL, M. K—SHASHI PRABHA ARYA: On almost regular spaces. Glasnik Mat. 24, 1969, 89—99. [23] STALLINGS, J.: Fixed point theorems for connectivity maps. Fundam. Math. 47, 1959, 249—263. [24] WILANSKI ALBERT: Topics in Functional Analysis. Springer-Verlag, Berlin 1967. Received November 28, 1975 Department oť Mathematics Baranas Hindu University Varanasi 221005. India Department oť Mathematics University oť Allahabad Allahabad. India 248 

Somewhat continuity and some other generalisations of continuity

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Somewhat continuity and some other generalisations of continuity

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Institute of Mathematics AS CR, v. v. i.